Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 265 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.What is the units digit of 3^33? (a) 9 (b) 1 (c) 7 (d) 3
2. 2.Which of these numbers is divisible by 4? (a) 1512 (b) 1517 (c) 1515 (d) 1510
3. 3.Which of these numbers is divisible by 3? (a) 559 (b) 553 (c) 555 (d) 557
4. 4.Which of these numbers is divisible by 11? (a) 9024 (b) 9017 (c) 9008 (d) 9009
5. 5.What is the units digit of 4^48? (a) 5 (b) 7 (c) 6 (d) 4
6. 6.What is the remainder when 319 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
7. 7.What is the units digit of 3^32? (a) 9 (b) 7 (c) 1 (d) 3
8. 8.What is the remainder when 128 is divided by 11? (a) 4 (b) 6 (c) 8 (d) 7
9. 9.What is the smallest number you must add to 367 to make it divisible by 10? (a) 2 (b) 3 (c) 7 (d) 4
10. 10.What is the units digit of 9^55? (a) 8 (b) 9 (c) 7 (d) 1
11. 11.What is the smallest number you must add to 588 to make it divisible by 9? (a) 3 (b) 5 (c) 6 (d) 7
12. 12.Which of these numbers is divisible by 11? (a) 3021 (b) 3010 (c) 3014 (d) 3034
13. 13.What is the remainder when 179 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 4? (a) 1730 (b) 1735 (c) 1732 (d) 1737
15. 15.What is the remainder when 299 is divided by 11? (a) 2 (b) 9 (c) 1 (d) 3
16. 16.What is the smallest number you must add to 435 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 7
17. 17.What is the remainder when 311 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
18. 18.What is the remainder when 477 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
19. 19.What is the smallest number you must add to 151 to make it divisible by 4? (a) 3 (b) 2 (c) 1
20. 20.Which of these numbers is divisible by 11? (a) 4675 (b) 4676 (c) 4674 (d) 4696
21. 21.What is the units digit of 8^59? (a) 2 (b) 8 (c) 4 (d) 6
22. 22.What is the units digit of 4^32? (a) 7 (b) 5 (c) 6 (d) 4
23. 23.Which of these numbers is divisible by 3? (a) 945 (b) 943 (c) 950 (d) 946
24. 24.Which of these numbers is divisible by 9? (a) 639 (b) 653 (c) 640 (d) 636
25. 25.What is the units digit of 2^30? (a) 4 (b) 6 (c) 8 (d) 2
26. 26.Which of these numbers is divisible by 6? (a) 1151 (b) 1142 (c) 1138 (d) 1140
27. 27.Which of these numbers is divisible by 3? (a) 991 (b) 990 (c) 994 (d) 989
28. 28.What is the smallest number you must add to 472 to make it divisible by 9? (a) 7 (b) 4 (c) 5 (d) 6
29. 29.What is the units digit of 9^33? (a) 7 (b) 9 (c) 8 (d) 1
30. 30.Which of these numbers is divisible by 6? (a) 1051 (b) 1050 (c) 1061 (d) 1049
31. 31.Which of these numbers is divisible by 11? (a) 3473 (b) 3479 (c) 3489 (d) 3476
32. 32.What is the smallest number you must add to 217 to make it divisible by 9? (a) 1 (b) 8 (c) 6 (d) 7
33. 33.What is the remainder when 804 is divided by 11? (a) 0 (b) 1 (c) 2 (d) 10
34. 34.What is the smallest number you must add to 249 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
35. 35.What is the units digit of 4^56? (a) 5 (b) 6 (c) 4 (d) 7
36. 36.What is the smallest number you must add to 693 to make it divisible by 12? (a) 9 (b) 2 (c) 3 (d) 4
37. 37.Which of these numbers is divisible by 8? (a) 1528 (b) 1527 (c) 1538 (d) 1531
38. 38.Which of these numbers is divisible by 9? (a) 1032 (b) 1037 (c) 1026 (d) 1023
39. 39.What is the smallest number you must add to 334 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 6
40. 40.Which of these numbers is divisible by 4? (a) 390 (b) 387 (c) 384 (d) 382

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